Two-element fθ lens used for micro-electro mechanical system (MEMS) laser scanning unit

ABSTRACT

A two-element f-θ lens used for a micro-electro mechanical system (MEMS) laser scanning unit includes a first lens and a second lens, the first lens is a positive refraction meniscus lens of which the convex surface is disposed on a side of a MEMS mirror, the second lens is a positive refraction meniscus lens of which the concave surface is disposed on the side of the MEMS mirror, at least one optical surface is an Aspherical surface in both main scanning direction and sub scanning direction, and satisfies special optical conditions. The two-element f-θ lens corrects the nonlinear relationship between scanned angle and time into the linear relationship between image spot distances and time. The two-element f-θ lens focuses the scan light to the target in the main scanning and sun scanning directions, such that the purpose of the scanning linearity effect and the high resolution scanning can be achieved.

FIELD OF THE INVENTION

The present invention relates to a two-element fθ lens used for amicro-electro mechanical system (MEMS) laser scanning unit (LSU), andmore particularly to a two-element fθ lens using an angular changevarying with time in a sinusoidal relation for correcting a MEMSreflecting mirror having a simple harmonic movement to achieve thescanning linearity effect by the laser scanning unit.

DESCRIPTION OF THE RELATED ART

At present, a laser scanning unit (LSU) used by a laser beam printer(LBP) controls a laser beam scanning by a high-speed rotating polygonmirror as disclosed in U.S. Pat. Nos. 7,079,171, 6,377,293 and 6,295,116or TW Pat No.1198966, and principles of those inventions are describedas the following: a semiconductor laser emits a laser beam through acollimator and an aperture to form parallel beams. After the parallelbeams pass through a cylindrical lens, the beams are focused at thewidth of the X-axis in the sub scanning direction and along a directionparallel to the Y-axis of the main scanning direction to form a lineimage and projected onto a high-speed rotating polygon mirror. Thepolygon mirror includes a plurality of continuous reflecting mirrorsdisposed substantially at or proximate to the focusing position of theline image. The polygon mirror is provided for controlling the directionof projecting the laser beam, so that when a plurality of continuousreflecting mirrors are rotated at high speed, the laser beam projectedonto a reflecting mirror can be extended in a direction parallel to themain scanning direction(Y-axis) at the same angular velocity anddeviated from and reflected onto a fθ linear scanning lens. The fθlinear scanning lens is installed next to the polygon mirror and may beeither a single-element lens structure (or a single scanning lens) or atwo-element lens structure. The function of this fθ linear scanning lensis to focus a laser beam reflected by the reflecting mirror of thepolygon mirror and projected onto the fθ lens into an oval spot that isprojected onto a photoreceptor (or a photoreceptor drum, which is animage surface) to achieve the requirement of the scanning linearity.However, the traditional laser scanning unit (LSU) still has thefollowing drawbacks in its practical use.

(1) The manufacture of the rotating polygon mirror incurs a high levelof difficulty and a high cost, and thus increasing the manufacturingcost of the LSU.

(2) The polygon mirror requires a function of a high-speed rotation(such as 40000 rpm) and a high precision, and thus a cylindrical lens isrequired and installed to the traditional LSU since the width of thegeneral polygon mirror along the Y-axis of the reflecting surface of themirror is very thin, so that the laser beam pass through the cylindricallens can be focused and concentrated into a line (or a spot on theY-axis) and projected onto the reflecting mirror of the polygon mirror.Such arrangement increases the number of components and also complicatesthe assembling operation procedure.

(3) The traditional polygon mirror requires a high-speed rotation (suchas 40000 rpm), and thus the noise level is raised. Furthermore, thepolygon mirror takes a longer time to be accelerated from a startingspeed to an operating speed, and thus increasing the booting time of thelaser scanning.

(4) In the fabrication of the traditional LSU, the central axis of alaser beam projected onto the reflecting mirror of the polygon mirror isnot aligned precisely with the central rotating axis of the polygonmirror, so that it is necessary to take the off axis deviation of thepolygon mirror into consideration for the design of the fθ lens, andthus increasing the difficulty of design and manufacturing the fθ lens.

In recent years, an oscillatory MEMS reflecting mirror is introduced toovercome the shortcomings of the traditional LSU assembly and replacethe laser beam scanning controlled by the traditional polygon mirror.The surface of a torsion oscillator of the MEMS reflecting mirrorcomprises a reflecting layer, and the reflecting layer is oscillated forreflecting the light and further for the scanning. In the future, sucharrangement will be applied in a laser scanning unit (LSU) of an imagingsystem, a scanner or a laser printer, and its scanning efficiency ishigher than the traditional rotating polygon mirror. As disclosed in theU.S. Pat. Nos. 6,844,951 and 6,956,597, at least one driving signal isgenerated, and its driving frequency approaches the resonant frequencyof a plurality of MEMS reflecting mirrors, and the driving signal drivesthe MEMS reflecting mirror to produce a scanning path. In U.S. Pat. Nos.7,064,876, 7,184,187, 7,190,499, 2006/0033021, 2007/0008401 and2006/0279826 or TW Pat No. 253133, or JP Pat. No. 2006-201350, a MEMSreflecting mirror installed between a collimator and a fθ lens of a LSUmodule replaces the traditional rotating polygon mirror for controllingthe projecting direction of a laser beam. The MEMS reflecting mirrorfeatures the advantages of small components, fast rotation, and lowmanufacturing cost. However, after the MEMS reflecting mirror is drivenby the received voltage for a simple harmonic with a sinusoidal relationof time and angular speed, and a laser beam projected on the MEMSreflecting mirror is reflected with a relation of reflecting angle 0(t)and time as followsθ(t)=θ_(s)·sin(2π·f·t)  (1)

wherein, f is the scanning frequency of the MEMS reflecting mirror andθ_(s) is the maximum scanning angle at a single side (symmetrical withthe optical Z axis) after the laser beam passes through the MEMSreflecting mirror.

In the same time interval Δt, the corresponding variation of thereflecting angle is not the same but decreasing, and thus constituting asinusoidal relation with time. In other words, the variation of thereflecting angle in the same time interval Δt isΔθ(t)=θ_(s)·(sin(2π·f·t₁)−sin(2π·f·t₂)), which constitutes a non-linearrelation with time. If the reflected light is projected onto the targetfrom a different angle, the distance from the spot will be different inthe same time interval due to the different angle.

Since the angle of the MEMS reflecting mirror situated at a peak and avalley of a sine wave varies with time, and the rotating movements if atraditional polygon mirror are at a constant angular speed, if atraditional fθ lens is installed on a laser scanning unit (LSU) of theMEMS reflecting mirror, the angle of the MEMS reflecting mirror producedby the sinusoidal relation varied with time cannot be corrected, so thatthe speed of laser beam projected on an image side will not be anuniform speed, and the image on the image side will be deviated.Therefore, the laser scanning unit or the MEMS laser scanning unit (MEMSLSU) composed of MEMS reflecting mirrors has a characteristic that afterthe laser beam is scanned by the MEMS reflecting mirror, scan lights atdifferent angles are formed in the same time. Thus, finding a way ofdeveloping a f0 lens (some prior art named as f-sin θ lens) for the MEMSlaser scanning unit to correct the scan lights, such that a correctimage will be projected onto the light, examples as, U.S. Pat. No.7,184,187 provided a polynomial surface for fθ lens to adjust theangular velocity variation in the main-scanning direction only. However,the laser light beam is essential an oval-like shape of the crosssection that corrects the scan lights in the main-scanning directiononly may not be achieve the accuracy requirement. Since, a fθ lens withmain-scanning direction correcting as well as sub-scanning directioncorrecting demands immediate attentions and feasible solutions.

SUMMARY OF THE INVENTION

The primary objective of the present invention is to overcome theshortcomings of the prior art by providing a two-element lens used for amicro-electro mechanical system (MEMS) laser scanning unit, whichcomprises a first lens in positive refraction meniscus shape having aconvex surface on a side of a MEMS mirror, and a second lens in apositive refraction meniscus lens having a concave surface on the sideof the MEMS mirror, counted from the MEMS reflecting mirror, forprojecting a scan light reflected by the MEMS reflecting mirror onto thecorrect image of a target to achieve a scanning linearity effectrequired by the laser scanning unit.

Another objective of the present invention is to provide a two-elementfθ lens used for a micro-electro mechanical system (MEMS) laser scanningunit for reducing the area of a spot projected onto the target toachieve the effect of improving the resolution.

A further object of the present invention is to provide a two-element fθlens used for a MEMS laser scanning unit, and the two-element fθ lenscan make a distortion correction to correct optical axis caused by thedeviation of the scan light resulting in the problems of an increaseddeviation of the main scanning direction and the sub scanning direction,and a change of a spot of a drum at the image into an oval like shape,and the two-element fθ lens can unify the size of each image spot toachieve the effect of enhancing the image quality.

Therefore, the two-element lens used for a micro-electro mechanicalsystem (MEMS) laser scanning unit of the invention is applicable for alight source comprising an emitting laser beam, wherein a resonantoscillation is used for reflecting the laser beam of the light sourceonto MEMS reflecting mirror of the scan light to form an image on thetarget. As to a laser printer, the target is generally a drum. The spotof the image forms a scan light after the laser beam is emitted from thelight source, scanned oscillatory by the MEMS reflecting mirror, andreflected by the MEMS reflecting mirror. After the angle and position ofthe scan light are corrected by the two-element fθ lens of theinvention, a spot may be formed on the drum. Since a photosensitiveagent is coated onto the drum, data can be printed out on a piece ofpaper by the sensing carbon powder centralized on the paper.

The two-element fθ lens of the invention comprises a first lens and asecond lens, counted from the MEMS reflecting mirror, wherein the firstlens includes a first optical surface and a second optical surface, thesecond lens includes a third optical surface and a fourth opticalsurface. These optical surface provided the functions of correcting thephenomenon of non-uniform speed scanning which results in decreasing orincreasing the distance between spots on an image surface of a MEMSreflecting mirror with a simple harmonic movement with time into aconstant speed scanning, so that the projection of a laser beam onto animage side can give a constant speed scanning, and unify the deviationof image formed on the drum which caused by a scan light in the mainscanning direction and the sub scanning direction deviated from theoptical axis, so as to make the correction to focus the scan light at atarget.

To make it easier for our examiner to understand the technicalcharacteristics and effects of the present invention, we use preferredembodiments and related drawings for the detailed description of thepresent invention as follows.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows optical paths of a two-element fθ lens of the presentinvention;

FIG. 2 shows a relation of scanning angle θ versus time t of the MEMSreflecting mirror;

FIG. 3 shows an optical path chart and numerals of a scan light passingthrough a first lens and a second lens;

FIG. 4 shows a spot area varied with a different projecting positionafter a scan light is projected onto a drum;

FIG. 5 shows the Y direction of Gaussian beam diameter of scanning lightemitted by fθ lens;

FIG. 6 shows an optical path chart of a scan light passing through afirst lens and a second lens;

FIG. 7 shows spots in accordance with a first preferred embodiment ofthe present invention;

FIG. 8 shows spots in accordance with a second preferred embodiment ofthe present invention;

FIG. 9 shows spots in accordance with a third preferred embodiment ofthe present invention;

FIG. 10 shows spots in accordance with a fourth preferred embodiment ofthe present invention; and

FIG. 11 shows spots in accordance with a fifth preferred embodiment ofthe present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1 for a schematic view of optical paths of atwo-element fθ lens used for micro-electro mechanical system (MEMS)laser unit in accordance with the present invention, the two-element fθlens used for the micro-electro mechanical system (MEMS) laser scanningunit comprises: a first lens 131 having a first optical surface 131 aand a second optical surface 131 b, and a second lens having a thirdoptical surface 132 a and a fourth optical surface 132 b. In FIG. 1, theMEMS laser scanning unit comprises a laser source 11, a MEMS reflectingmirror 10, a cylindrical lens 16, two photoelectric sensors 14 a, 14 band a light sensing target. In FIG. 1 the target is implemented by adrum 15. After a beam 111 produced by the light laser source 11 ispassed through a cylindrical lens 16, the beam 111 is projected onto theMEMS reflecting mirror 10. The MEMS reflecting mirror 10 generates aresonant oscillation to reflect the beam 111 into scan lights 113 a, 113b, 114 a, 114 b, 115 a, 115 b at different time frames along thedirection of Z, wherein the scan lights 113 a, 113 b, 114 a, 114 b, 115a, 115 b are projected in a X direction which is called a sub scanningdirection, and projected in a Y direction which is called a mainscanning direction, and the maximum scanning angle of the MEMSreflecting mirror 10 is θc.

Since the MEMS reflecting mirror 10 comes with a simple harmonicmovement, and the angle of movement shows a sinusoidal change with timeas shown in FIG. 2, therefore the angle and time of reflecting the scanlight are in a non-linear relation. The swinging angle of the MEMSreflecting mirror 10 has a wave peak a-a′ and a wave valley b-b′ asshown in the figure, and its swinging angle is significantly smallerthan the wave sections a-b an a′-b′, and this non-uniform angular speedmay cause an image deviation easily produced on the drum 15 by the scanlight. Therefore, photoelectric sensors 14 a, 14 b are installed at theangle ±θp within the range below the maximum scanning angle ±θc of theMEMS reflecting mirror 10 and the laser beam 111 starts to be reflectedby the MEMS reflecting mirror 10 at the wave peak as shown in FIG. 2,which is equivalent to the scan light 115 a as shown in FIG. 1. If thephotoelectric sensor 14 a detects a scanned beam, it means that the MEMSreflecting mirror 10 swings to an angle of ±θp, which is equivalent tothe scan light 114 a as shown in FIG. 1. If the MEMS reflecting mirror10 scans point “a” at an angle variation as shown in FIG. 2, such pointis equivalent to the position of the scan light 113 a. Now, the lasersource 11 is controlled to start emitting the laser beam 111. When thepoint “b” as shown in FIG. 2 is scanned, such point is equivalent to theposition of the scan light 113 b (which is equivalent to the laser beam111 emitted by the laser source 11 a within an angle of ±θn). When theMEMS reflecting mirror 10 swings in an opposite direction to a wavesection a′-b′, the laser source 11 is controlled to start emitting thelaser beam 111 to complete a cycle.

Referring to FIG. 3 for an optical path chart of a scan light passingthrough a first lens and a second lens, in which ±θn is a valid scanningangle. If the MEMS reflecting mirror 10 is swung to the angle of ±θn,the laser source 11 starts emitting the desired scanning laser beam 111which is reflected into a scan light by the MEMS reflecting mirror 10,and the scan light is passed through the first lens 131 and refracted bythe first optical surface and the second optical surface of the firstlens 131, and the scan light reflected by the MEMS reflecting mirror 10with a none-linear relation between distance and time is converted intoa scan light with a linear relation between distance and time. After thescan light is passed through the first lens 131 and the second lens 132,the focusing effect of the first optical surface 131 a, the secondoptical surface 131 b, the third optical surface 132 a and the fourthoptical surface 132 b of the first lens 131 and the second lens 132 andthe interval of each optical surface can focus the scan light at thedrum 15 and form a column of spots 2 on the drum 15, and the distancebetween the farthest two spots projected on the drum 15 is called aneffective scanning windows 3, wherein along the optical axis Z, d₁ isthe distance between the MEMS reflecting mirror 10 and the first opticalsurface, d₂ is the distance between the first optical surface and thesecond optical surface, d₃ is the distance between the second opticalsurface, R₂ and the third optical surface, d₄ is the distance betweenthe third optical surface and the fourth optical surface R₄, d₅ is thedistance between the fourth optical surface and the drum, R₁ is theradius of curvature of the first optical surface, R₂ is the radius ofcurvature of the second optical surface, R₃ is the radius of curvatureof the third optical surface, R₄ is the radius of curvature of thefourth optical surface in the optical axis.

Referring to FIG. 4 for a spot area varied with a different projectingposition after a scan light is projected onto a drum, if the scan light113a is projected in a direction along the optical axis Z and onto thedrum 15 by the first lens 131 and the second lens 132, the incidentangles of the first lens 131 and the second lens 132 are zero, and thusthe deviation of the main scanning direction is minimum (said zero), andthe image at the spot 2 a on the drum 15 is in an inferenced circle-likeshape (same shape as laser light beam). After the scan light 113 b and113 c is projected on the drum 15 by the first lens 131 and the secondlens 132, the incident angle of the first lens 131 and the second lens132 with respect to the optical axis is non-zero, and the deviation ofthe main scanning direction is non-zero, and thus the projectiondistance of the main scanning direction is longer than the spot formedby the scan light 111 a is also bigger. Not only has the phenomenonexisted in the main scanning direction but also in the sub scanningdirection. Therefore, the image at the spot 2 b, 2 c on the drum 15 isin an oval-like shape, and the area of 2 b, 2 c is greater than the areaof 2a. Denoted S_(a0) and S_(b0) are the lengths of spots of the scanlight in the main scanning direction (Y direction) and the sub scanningdirection (X direction) on a reflecting surface of the MEMS reflectingmirror 10, and G_(a0) and G_(b0) are the Gaussian beam diameter ofscanning light emitted by fθ lens 13 at the intensity is 13.5% ofmaximum intensity on Y direction and the X direction, illustrated byFIG. 5. In FIG. 5, only Y direction Gaussian beam is shown. Thetwo-element fθ lens of the invention can control the spot size in themain scanning direction within a limited range by the distortioncorrection of the fθ lens 13 and correct the spot size in the subscanning direction by the distortion correction of the first lens 131and the second 132 of the two-element fθ lens 13, such that the spotsize is controlled within a limited range, and the distribution of thespot size (or the ratio of largest spots and smallest spots) iscontrolled within an appropriate range in compliance with the requiredresolution.

To achieve the forgoing effects, the two-element fθ lens of theinvention comes with a first lens having a first optical surface and asecond optical lens having a third optical surface and a fourth opticalsurface of with a spherical surface or an aspherical surface. If theAspherical surface is adopted, the aspherical surface is designed withthe following equations (2) or (3)

1. Anamorphic Equation

$\begin{matrix}{Z = {\frac{{({Cx})X^{2}} + {({Cy})Y^{2}}}{1 + \sqrt{1 - {\left( {1 + {Kx}} \right)({Cx})^{2}X^{2}} - {\left( {1 + {Ky}} \right)({Cy})^{2}Y^{2}}}} + {A_{R}\left\lbrack {{\left( {1 - A_{P}} \right)X^{2}} + {\left( {1 + A_{P}} \right)Y^{2}}} \right\rbrack}^{2} + {B_{R}\left\lbrack {{\left( {1 - B_{P}} \right)X^{2}} + {\left( {1 + B_{P}} \right)Y^{2}}} \right\rbrack}^{3} + {C_{R}\left\lbrack {{\left( {1 - C_{P}} \right)X^{2}} + {\left( {1 + C_{P}} \right)Y^{2}}} \right\rbrack}^{4} + {D_{R}\left\lbrack {{\left( {1 - D_{P}} \right)X^{2}} + {\left( {1 + D_{P}} \right)Y^{2}}} \right\rbrack}^{5}}} & (2)\end{matrix}$

where, Z is the sag of any point on the surface parallel to the Z-axis,C_(x) and C_(y) are curvatures in the X direction and the Y directionrespectively, K_(x) and K_(y) are the conic coefficients in the Xdirection and the Y direction respectively and correspond to eccentriccity in the same way as conic coefficient for the Aspherical surfacetype, A_(R), B_(R), C_(R) and D_(R) are deformations from the coniccoefficient of rotationally symmetric portions of the fourth order, thesixth order, the eighth and the tenth order respectively, and A_(P),B_(P), C_(P) and D_(P) are deformation from the conic coefficient ofnon-rotationally symmetric components to the fourth order, the sixthorder, the eight order and the tenth order respectively. This reduces toAspherical surface type when C_(x)=C_(y), K_(x)=K_(y) andΛ_(p)=B_(p)=C_(p)=D_(p)=0.

2. Toric Equation

$\begin{matrix}{\mspace{79mu}{{Z = {{Zy} + \frac{({Cxy})X^{2}}{1 + \sqrt{1 - {({Cxy})^{2}X^{2}}}}}}\mspace{79mu}{{Cxy} = \frac{1}{\left( {1/{Cx}} \right) - {Zy}}}{{Zy} = {\frac{({Cy})Y^{2}}{1 + \sqrt{1 - {\left( {1 + {Ky}} \right)({Cy})^{2}Y^{2}}}} + {B_{4}Y^{4}} + {B_{6}Y^{6}} + {B_{8}Y^{8}} + {B_{10}Y^{10}}}}}} & (3)\end{matrix}$

where, Z is the sag of any point on the surface parallel to the Z-axis;C_(y) and C_(x) are curvatures in the X direction and the Y directionrespectively, K_(y) is a conic coefficient in the Y direction, B₄, B₆,B₈ and B₁₀ are deformations from the conic coefficient to the fourth,sixth, eight and tenth order respectively. When C_(x)=C_(y) andK_(y)=A_(p)=B_(p)=C_(p)=D_(p)=0 is reduced to a single sphericalsurface.

To unify the scan speed of the scan light projected onto the image ofthe target, the invention adopts two equal time interval and an equaldistance between two spots, and the two-element fθ lens of the inventioncan correct the emergence angle of the scan light between the scan light113 a to the scan light 113 b, so that the first lens 131 and the secondlens 132 corrects the emergence angle of the scan light to produce twoscan lights at the same time interval. After the emergence angle iscorrected, the distance between any two spots formed on the drum 15 ofthe image is equal. Further, after the laser beam 111 is reflected bythe MEMS reflected mirror 10, the spot is diverged and becomes larger.After the scan light is passed through the distance from the MEMSreflecting mirror 10 to the drum 15, the spot becomes larger. Sucharrangement is incompliance with the actual required resolution. Thetwo-element fθ lens of the invention further focuses from the scan light113 a to the scan light 113 b reflected by the MEMS reflecting mirror 10at the drum 15 of the image to from a smaller spot in the main scanningand sub scanning directions. The two-element fθ lens of the inventionfurther uniforms the spot size of the image on the drum 15 (to limitspot size in a range to comply with the required resolution) for thebest condition.

The two-element fθ lens comprises a first lens 131 and a second lens132, counted from the MEMS reflecting mirror 10, and the first lens is apositive refraction meniscus lens of which the convex surface isdisposed on a side of a MEMS mirror, the second lens is a positiverefraction meniscus lens of which the concave surface is disposed on theside of the MEMS mirror, wherein the first lens 131 includes a firstoptical surface 131 a and a second optical surface 131 b for convertinga scan spot with a non-linear relation between angle and time andreflected by the MEMS reflecting mirror 10 into a scan spot with alinear relation between distance and time; and the second lens 132includes a third optical surface 132 a and a fourth optical surface 132b for correcting the focus of the scan light of the first lens 131 ontotarget; such that the two-element fθ lens projects a scan lightreflected by the MEMS reflecting mirror 10 onto the image of the drum15. The first optical surface 131 a, the second optical surface 131 b,the third optical surface 132 a and the fourth optical surface 132 b areoptical surfaces composed of at least one Aspherical surface in the mainscanning direction. The first optical surface 131 a and the secondoptical surface 131 b are optical surfaces composed of at least oneaspherical surface in the sub scanning direction. Further, the assemblyof the first lens 131 and the second lens 132 of the two-element fθ lensin accordance with the present invention has an optical effect in themain scanning direction that satisfies the conditions of Equation (4)and (5)

$\begin{matrix}{0.1 < \frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}} < 0.8} & (4) \\{0.2 < \frac{d_{5}}{f_{{(2)}Y}} < 0.8} & (5) \\{0.3 < {{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right)}} < 0.6} & (6)\end{matrix}$and the sub scanning direction satisfies the conditions of equation (7)

$\begin{matrix}{0.8 < {{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}} < 1.6} & (7)\end{matrix}$

where, f_((1)Y) is the focal length of the first lens 131 in the mainscanning direction, f_((2)Y) is the focal length of the second lens 132in the main scanning direction, d₃ is the distance between an opticalsurface on a target side of the first lens 131 when θ=0° and an opticalsurface on the MEMS reflecting mirror side of the second lens 132, d₄ isthe thickness of the second lens when θ=0°, d₅ is the distance betweenan optical surface on a target side of the second lens 132 when θ=0° andthe target, f_((1)X) is the focal length of the first lens in the subscanning direction, f_((2)X) is the focal length of the second lens inthe sub scanning direction, f_(s) is the combined focal length of thetwo-element fθ lens, R_(ix) is the radius of curvature of the i-thoptical surface in the X direction; and n_(d1) and n_(d2) are therefraction indexes of the first lens and the second lens 13respectively.

Further, the spot uniformity produced by the two-element fθ lens of theinvention can be indicated by the ratio δ of the largest spot and thesmallest spot size that satisfies the conditions of Equation (8):

$\begin{matrix}{{0.4 < \delta} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}} & (8)\end{matrix}$

The resolution produced by the two-element fθ lens of the invention canbe indicated by the ratio η_(max) of the largest spot on the drum 15formed by the scan light on the reflecting surface of the MEMSreflecting mirror 10 (or the ratio of scanning light of maximum spot)and the ratio η_(min) of the smallest spot formed by the scan light onthe reflecting surface of the MEMS reflecting mirror 10 (or the ratio ofscanning light of minimum spot), and the ratios satisfy the conditionsof Equations (9) and (10)

$\begin{matrix}{\eta_{\max} = {\frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}} & (9) \\{\eta_{\min} = {\frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}} & (10)\end{matrix}$

where, S_(a) and S_(b) are the lengths of any one spot of the scan lightformed on the drum in the main scanning direction and the sub scanningdirection, δ is the ratio of the smallest spot and the largest spot onthe drum 15, S_(a0) and S_(b0) are the lengths of the spots of the scanlight on the reflecting surface of the MEMS reflecting mirror 10 in themain scanning direction and the sub scanning direction respectively.

To make it easier for our examiner to understand the structure andtechnical characteristics of the present invention, we use the preferredembodiments accompanied with related drawings for the detaileddescription of the present invention as follows.

The following preferred embodiments of the invention disclose atwo-element fθ lens used for a micro-electro mechanical system (MEMS)laser scanning unit by using major elements for the illustration, andthus the preferred embodiments can be applied in a MEMS laser scanningunit including but not limited to the two-element fθ lens withcomponents illustrated in the embodiments only, but any otherequivalents are intended to be covered in the scope of the presentinvention. In other words, any variation and modification of thetwo-element fθ lens used for a micro-electro mechanical system (MEMS)laser scanning unit can be made by the persons skilled in the art. Forexample, the radius of curvature of the first lens and the second lens,the design of the shape, the selected material and the distance can beadjusted without any particular limitation.

In a first preferred embodiment, the two-element fθ lens comprises afirst lens and a second lens. The first lens is a positive refractionmeniscus lens of which the convex surface is disposed on a side of aMEMS mirror, the second lens is a positive refraction meniscus lens ofwhich the concave surface is disposed on the side of the MEMS mirror,and a first optical surface and a second optical surface of the firstlens, a third optical surface and a fourth optical surface of the secondlens are all Aspherical surfaces designed in accordance with theEquation (2), and the optical characteristics and the Aspherical surfaceparameters are listed in Tables 1 and 2.

TABLE 1 Optical Characteristics of fθ lens for First PreferredEmbodiment nd, refraction optical surface radius (mm) d, thickness (mm)index MEMS Reflection R0 ∞ 15.63 1 lens 1 1.527 R1 (Anamorphic) R1x*70.866 9.33 R1y* 68.513 R2 (Anamorphic) R2x* −18.363 12.46 R2y* 148.516lens 2 1.527 R3 (Anamorphic) R3x* 45.141 10.04 R3y* −343.393 R4(Anamorphic) R4x* −61.261 113.06 R4y* −95.690 drum R5 ∞ 0.00 *Asphericalsurface

TABLE 2 Parameters of Aspherical Surface of Optical Surface Parameterfor First Preferred Embodiment Anamorphic equation coefficient 4th Order6th Order 8th Order 10th Order Ky, Conic Coefficient CoefficientCoefficient Coefficient optical surface Coefficient (AR) (BR) (CR) (DR)R1* 2.5171E+00 0.000000 0.000000 0.000000 0.000000 R2* −1.0000E+01−1.0616E−06 2.4812E−09 0.0000E+00 0.0000E+00 R3* 1.3560E+02 −4.0354E−05−5.6870E−12 0.0000E+00 0.0000E+00 R4* −8.6868E+00 3.5599E−07 9.6087E−100.0000E+00 0.0000E+00 4th Order 6th Order 8th Order 10th Order Kx, ConicCoefficient Coefficient Coefficient Coefficient Coefficient (AP) (BP)(CP) (DP) R1* 10.746917 −0.443978 0.000000 0.000000 0.000000 R2*4.5725E+00 1.7785E−01 0.0000E+00 0.0000E+00 0.0000E+00 R3* 4.4379E+00−9.0564E−01 0.0000E+00 0.0000E+00 0.0000E+00 R4* −2.0030E+01 0.0000E+000.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=231.883, f_((2)Y)=248.128,f_(sX)=23.211, f_(sY)=128.531 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=12.90 and S_(b0)=4618.85(μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 3. The maximum diameter (μm) ofgeometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 4. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 7, where the diameter of unity circle is 0.05 mm.

TABLE 3 Conditions for first Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.5846$\frac{d_{5}}{f_{{(2)}Y}}$ 0.4556${Main}\mspace{14mu}{scanning}\mspace{14mu}{direction}{\;\;}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.5650${Sub}\mspace{14mu}{scanning}\mspace{14mu}{direction}\mspace{11mu}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}}$0.9616$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.4636$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0693$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0321

TABLE 4 The maximum diameter (μm) of light spot on the drum Y −107.460−96.078 −84.297 −96.078 −60.291 −48.227 −36.161 −24.100 0.000 Maxdiameter 6.64E−03 6.83E−03 8.06E−03 6.44E−03 5.45E−03 6.40E−03 6.56E−035.43E−03 3.03E−03

In a second preferred embodiment, the two-element fθ lens comprises afirst lens and a second lens. The first lens is a positive refractionmeniscus lens of which the convex surface is disposed on a side of aMEMS mirror, the second lens is a positive refraction meniscus lens ofwhich the concave surface is disposed on the side of the MEMS mirror,and a first optical surface and a second optical surface of the firstlens, a third optical surface and a fourth optical surface of the secondlens are all Aspherical surfaces designed in accordance with theEquation (2), and the optical characteristics and the Aspherical surfaceparameters are listed in Tables 5 and 6.

TABLE 5 Optical characteristics of fθ lens for Second PreferredEmbodiment. nd, refraction optical surface radius (mm) d, thickness (mm)index MEMS Reflection R0 ∞ 15.89 1 lens 1 1.527 R1 (Anamorphic) R1x*167.228 10.00 R1y* 69.132 R2 (Anamorphic) R2x* −20.721 12.68 R2y*152.029 lens 2 1.527 R3 (Anamorphic) R3x* 46.359 10.38 R3y* −346.651 R4(Anamorphic) R4x* −44.659 112.48 R4y* −96.030 drum R5 ∞ 0.00 *Asphericalsurface

TABLE 6 Parameters of Aspherical Surface of Optical Surface Parameterfor Second Preferred Embodiment Anamorphic equation Coefficient 4thOrder 6th Order 8th Order 10th Order Ky, Conic Coefficient CoefficientCoefficient Coefficient optical surface Coefficient (AR) (BR) (CR) (DR)R1* 2.2690E+00 7.4099E−07 8.3470E−10 0.0000E+00 0.0000E+00 R2*−6.9331E+00 −9.9351E−07 2.4655E−09 0.0000E+00 0.0000E+00 R3* 1.3189E+02−2.3733E−05 1.8610E−10 0.0000E+00 0.0000E+00 R4* −9.1332E+00 4.8026E−081.0017E−09 0.0000E+00 0.0000E+00 4th Order 6th Order 8th Order 10thOrder Kx, Conic Coefficient Coefficient Coefficient CoefficientCoefficient (AP) (BP) (CP) (DP) R1* 7.5867E+01 −2.0141E−01 0.0000E+000.0000E+00 0.0000E+00 R2* 3.3010E+00 8.8174E−02 0.0000E+00 0.0000E+000.0000E+00 R3* 5.3248E+00 −8.5112E−01 0.0000E+00 0.0000E+00 0.0000E+00R4* −7.8178E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=230.847, f_((2)Y)=248.37,f_(sX)=25.189, f_(sY)=128.63 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=12.90 and S_(b0)=4618.85(μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 7. The maximum diameter (μm) ofgeometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 8. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 8, where the diameter of unity circle is 0.05 mm.

TABLE 7 Conditions for Second Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.5871$\frac{d_{5}}{f_{{(2)}Y}}$ 0.4528${Main}\mspace{14mu}{scanning}\mspace{14mu}{direction}{\;\;}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.5665${Sub}\mspace{14mu}{scanning}\mspace{14mu}{direction}\mspace{11mu}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}}$1.1616$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.4691$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0604$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0283

TABLE 8 The maximum diameter (μm) of geometric spot on the drum Y−107.460 −96.078 −84.301 −96.078 −60.302 −48.240 −36.174 −24.112 0.000Max diameter 5.22E−03 3.82E−03 5.54E−03 4.92E−03 4.18E−03 4.22E−034.47E−03 3.99E−03 2.51E−03

In a third preferred embodiment, the two-element fθ lens comprises afirst lens and a second lens. The first lens is a positive refractionmeniscus lens of which the convex surface is disposed on a side of aMEMS mirror, the second lens is a positive refraction meniscus lens ofwhich the concave surface is disposed on the side of the MEMS mirror,and a first optical surface and a second optical surface of the firstlens, a third optical surface are all Aspherical surfaces designed inaccordance with the Equation (3), and a fourth optical surface of thesecond lens is an Aspherical surfaces designed in accordance with theEquation (2), and the optical characteristics and the Aspherical surfaceparameters are listed in Tables 9 and 10.

TABLE 9 Optical Characteristics of fθ Lens for Third PreferredEmbodiment nd, refraction optical surface radius (mm) d, thickness (mm)index MEMS Reflection R0 ∞ 15.00 1 Lens 1 1.527 R1 (Anamorphic) R1x*213.473 9.68 R1y* 71.710 R2 (Anamorphic) R2x* −20.940 14.30 R2y* 178.375Lens 2 1.527 R3 (Anamorphic) R3x* 44.933 12.00 R3y* −309.566 R4 (YToric) R4x −40.567 110.88 R4y* −95.577 drum R5 ∞ 0.00 *Asphericalsurface

TABLE 10 Parameters of Aspherical Surface for Third Preferred EmbodimentToric equation Coefficient 4th Order 6th Order 8th Order 10th Order Ky,Conic Coefficient Coefficient Coefficient Coefficient optical surfaceCoefficient (B4) (B6) (B8) (B10) R4* −9.6378E+00 0.0000E+00 0.0000E+000.0000E+00 0.0000E+00 Anamorphic equation Coefficient 4th Order 6thOrder 8th Order 10th Order Ky, Conic Coefficient Coefficient CoefficientCoefficient optical surface Coefficient (AR) (BR) (CR) (DR) R1*1.7853E+00 8.0497E−07 2.8518E−10 0.0000E+00 0.0000E+00 R2* −8.4076E+00−8.5575E−07 1.9710E−09 0.0000E+00 0.0000E+00 R3* 1.0000E+01 −1.9212E−053.6072E−10 0.0000E+00 0.0000E+00 4th Order 6th Order 8th Order 10thOrder Kx, Conic Coefficient Coefficient Coefficient CoefficientCoefficient (AP) (BP) (CP) (DP) R1* 1.8623E+02 −1.6418E−01 0.0000E+000.0000E+00 0.0000E+00 R2* 7.3392E+00 9.9473E−03 0.0000E+00 0.0000E+000.0000E+00 R3* −1.0000E+01 −8.6873E−01 0.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=128.339, f_((2)Y)=257.258,f_(sX)=26.0, f_(sY)=128.339(mm), so that the scan light can be convertedinto a scan spot with a linear relation of distance and time, and thespots with spot 3 S_(a0)=12.90 and S_(b0)=4618.85(μm) on the MEMSreflecting mirror 10 are scanned into scan lights and focused on thedrum 15 to form a smaller spot 6 and satisfy the conditions of Equations(4) to (10) as listed in Table 11. The maximum diameter (μm) ofgeometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 12. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 9, where the diameter of unity circle is 0.05 mm.

TABLE 11 Conditions for Third Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.6220$\frac{d_{5}}{f_{{(2)}Y}}$ 0.4310${Main}\mspace{14mu}{scanning}\mspace{14mu}{direction}{\;\;}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.5650${Sub}\mspace{14mu}{scanning}\mspace{14mu}{direction}\mspace{11mu}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}}$1.2719$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.4432$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0618$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0274

TABLE 12 The maximum diameter (μm) of geometric spot on the drum Y−108.245 −96.441 −84.420 −96.441 −60.230 −48.149 −36.092 −24.053 0.000Max diameter 5.99E−03 5.45E−03 3.76E−03 4.95E−03 5.04E−03 4.96E−034.54E−03 2.79E−03 4.53E−03

In a fourth preferred embodiment, the two-element fθ lens comprises afirst lens and a second lens. The first lens is a positive refractionmeniscus lens of which the convex surface is disposed on a side of aMEMS mirror, the second lens is a positive refraction meniscus lens ofwhich the concave surface is disposed on the side of the MEMS mirror,and a first optical surface and a second optical surface of the firstlens, a third optical surface and a fourth of the second lens are allAspherical surfaces designed in accordance with the Equation (2), andthe optical characteristics and the Aspherical surface parameters arelisted in Tables 13 and 14.

TABLE 13 Optical Characteristics of fθ lens for Fourth PreferredEmbodiment nd, refraction optical surface radius (mm) d, thickness (mm)index MEMS Reflection R0 ∞ 20.00 1 Lens 1 1.527 R1 (Anamorphic) R1x167.116 10.00 R1y* 70.438 R2 (Anamorphic) R2x* −22.346 13.41 R2y*156.952 Lens 2 1.527 R3 (Anamorphic) R3x* 59.818 12.00 R3y* −360.069 R4(Anamorphic) R4x* −49.577 111.87 R4y* −95.956 drum R5 ∞ 0.00 *Asphericalsurface

TABLE 14 Parameters of Aspherical Surface of Optical Surface for FourthPreferred Embodiment Anamorphic equation Coefficient 4th Order 6th Order8th Order 10th Order Ky, Conic Coefficient Coefficient CoefficientCoefficient optical surface Coefficient (AR) (BR) (CR) (DR) R1*2.2121E+00 6.1115E−07 4.6244E−10 0.0000E+00 0.0000E+00 R2* −2.8575E+00−1.1382E−06 1.8698E−09 0.0000E+00 0.0000E+00 R3* 1.2241E+02 −1.7276E−055.7782E−10 0.0000E+00 0.0000E+00 R4* −6.7720E+00 9.5679E−08 1.0726E−090.0000E+00 0.0000E+00 4th Order 6th Order 8th Order 10th Order Kx, ConicCoefficient Coefficient Coefficient Coefficient Coefficient (AP) (BP)(CP) (DP) R1* 4.7842E+01 −3.7954E−01 0.0000E+00 0.0000E+00 0.0000E+00R2* 2.4636E+00 −2.7582E−02 0.0000E+00 0.0000E+00 0.0000E+00 R3*7.6921E+00 −8.3511E−01 0.0000E+00 0.0000E+00 0.0000E+00 R4* −6.5777E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=233.066, f_((2)Y)=244.281,f_(sX)=27.931, f_(sY)=128.819 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=12.90 and S_(b0)=4618.85(μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 15. The maximum diameter (μm)of geometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 16. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 10, where the diameter of unity circle is 0.05 mm.

TABLE 15 Conditions for Fourth Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.5890$\frac{d_{5}}{f_{{(2)}Y}}$ 0.4580${Main}\mspace{14mu}{scanning}\mspace{14mu}{direction}{\;\;}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.5691${Sub}\mspace{14mu}{scanning}\mspace{14mu}{direction}\mspace{11mu}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}}$1.0810$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.4699$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0546$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0256

TABLE 16 The maximum diameter(μm) of geometric spot on target drum Y−107.455 −95.983 −84.201 −95.983 −60.245 −48.203 −36.159 −24.114 0.000Max diameter 7.53E−03 5.04E−03 4.90E−03 5.40E−03 6.89E−03 3.88E−033.44E−03 2.87E−03 4.92E−03

In a fifth preferred embodiment, the two-element fθ lens comprises afirst lens and a second lens. The first lens is a positive refractionmeniscus lens of which the convex surface is disposed on a side of aMEMS mirror, the second lens is a positive refraction meniscus lens ofwhich the concave surface is disposed on the side of the MEMS mirror, afirst optical surface and a second optical surface of the first lens, athird optical surface and a fourth optical surface of the second lensare all Aspherical surfaces designed in accordance with the Equation(2), and the optical characteristics and the Aspherical surfaceparameters are listed in Tables 17 and 18.

TABLE 17 Optical Characteristics of fθ Lens for Fifth PreferredEmbodiment nd, refraction optical surface radius (mm) d, thickness (mm)index MEMS Reflection R0 ∞ 20.00 1 Lens 1 1.527 R1 (Anamorphic) R1x*845.264 10.00 R1y* 78.384 R2 (Anamorphic) R2x* −32.166 20.00 R2y*224.025 Lens 2 1.527 R3 (Anamorphic) R3x* 61.059 11.26 R3y* −392.748 R4(Anamorphic) R4x* −34.884 107.97 R4y* −99.006 drum R5 ∞ 0.00 *Asphericalsurface

TABLE 18 Parameters of Aspherical Surface of Optical Surface for FifthPreferred Embodiment Anamorphic equation Coefficient 4th Order 6th Order8th Order 10th Order Ky, Conic Coefficient Coefficient CoefficientCoefficient optical surface Coefficient (AR) (BR) (CR) (DR) R1* 9.3832E−01 0.000000 0.000000 0.000000 0.000000 R2* −5.7249E+00−8.5402E−07 7.1261E−10 0.0000E+00 0.0000E+00 R3*  1.0287E+02 −7.8598E−061.2970E−10 0.0000E+00 0.0000E+00 R4* −9.1046E+00 −2.3728E−07 6.8959E−100.0000E+00 0.0000E+00 4th Order 6th Order 8th Order 10th Order Kx, ConicCoefficient Coefficient Coefficient Coefficient optical surfaceCoefficient (AP) (BP) (CP) (DP) R1* 1532.275398 −0.347992 0.0000000.000000 0.000000 R2*  6.9270E+00 −3.0389E−02 0.0000E+00 0.0000E+000.0000E+00 R3*  6.7524E+00 −8.6441E−01 0.0000E+00 0.0000E+00 0.0000E+00R4* −1.5624E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=233.383, f_((2)Y)=247.789,f_(sX)=33.301, f_(sY)=128.847 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=12.90 and S_(b0)=4618.85(μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 19. The maximum diameter (μm)of geometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 20. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 11, where the diameter of unity circle is 0.05 mm.

TABLE 19 Conditions for Fifth Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.6233$\frac{d_{5}}{f_{{(2)}Y}}$ 0.4357${Main}\mspace{14mu}{scanning}\mspace{14mu}{direction}{\;\;}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.5753${Sub}\mspace{14mu}{scanning}\mspace{14mu}{direction}\mspace{11mu}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}}$1.5323$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.4319$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0439$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0190

TABLE 20 The maximum diameter (μm) of geometric spot on target drum Y−108.537 −96.503 −84.379 −96.503 −60.177 −48.120 −36.081 −24.050 0.000Max diameter 2.04E−03 1.77E−03 1.62E−03 2.06E−03 1.24E−03 1.18E−031.30E−03 1.69E−03 9.22E−04

In view of the aforementioned preferred embodiments, the presentinvention at least has the following effects:

-   (1) With the two-element fθ lens of the invention, the scanning is    corrected the phenomenon of non-uniform speed which results in    decreasing or increasing the distance between spots on an image    surface of a MEMS reflecting mirror with a simple harmonic movement    with time into a constant speed scanning, so that the laser beam at    the image side is projected for a uniform speed scanning and an    equal distance between any two adjacent spots can be achieved for    the image on a target.-   (2) With the two-element fθ lens of the invention, the distortion    correction is provided for correcting the main scanning direction    and sub scanning direction of the scan light, so that the image size    of the spot focused at the target can be decreased.-   (3) With the two-element fθ lens of the invention, the distortion    correction is provided for correcting the main scanning direction    and the sub scanning direction of the scan light, so as to focus the    spot size focused and imaged at the target.

1. A two-element fθ lens used for a micro-electro mechanical system(MEMS) laser scanning unit, said MEMS laser scanning unit comprising alight source for emitting laser beam, a MEMS reflecting mirror forreflecting said laser beam emitted by said light source into a scanninglight by resonant oscillation, and a target provided for sensing light,said two-element fθ lens being disposed between said target and saidMEMS reflecting mirror, said two-element fθ lens comprising: a firstlens, in a positive refraction meniscus shape, and having a convexsurface toward said MEMS reflecting mirror; and a second lens, in apositive refraction meniscus shape, and having a concave surface towardsaid MEMS reflecting mirror, located between said first lens and saidtarget; wherein, said first lens included a first optical surface and asecond optical surface, at least one of said optical surfaces is anaspherical surface in both main scanning direction and sub scanningdirection of said MEMS laser scanning unit; wherein, said second lensincluded a third optical surface and a fourth optical surface, at leastone of said optical surfaces is an aspherical surface in both mainscanning direction and sub scanning direction of said MEMS laserscanning unit; wherein, said two-element fθ lens converts the non-linearrelation of reflecting angle with time of said scanning light into alinear relation between the distance of the scan spot with time andfocusing the scanning light to form an image at said target.
 2. Thetwo-element fθ lens of claim 1, wherein the main scanning directionsatisfies the conditions of:${0.1 < \frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}} < 0.8};$${0.2 < \frac{d_{5}}{f_{{(2)}Y}} < 0.8};$ wherein, f_((1)Y) is the focallength of the first lens in the main scanning direction, and f_((2)Y) isthe focal length of the second lens in the main scanning direction, andd₃ is the distance from the second optical surface to the third opticalsurface on the optical axis Z, and d₄ is the thickness of the secondlens along the optical axis Z, and d₅ is the distance from the fourthoptical surface to the target side on the optical axis Z.
 3. Thetwo-element fθ lens of claim 1, further satisfying the conditions of: inthe main scanning direction${0.3 < {{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right)}} < 0.6};$and in the sub scanning direction$0.8 < {{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}} < 1.6$wherein, f_((1)Y) and f_((1)X) are the focal lengths of the first lensin the main scanning direction and the sub scanning directionrespectively, and f_((2)Y) and f_((2)X) are the focal lengths of thesecond lens in the main scanning direction and the sub scanningdirection respectively, f_(s) is a combined focal length of thetwo-element fθ lens, and R_(ix) is the radius of curvature of the i-thoptical surface in the X direction; and n_(d1) and n_(d2) are refractionindexes of the first lens and the second lens respectively.
 4. Thetwo-element fθ lens of claim 1, wherein the ratio of the largest spotand the smallest spot size satisfies the conditions of:${{0.4 < \delta} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}};$wherein, S_(a) and S_(b) are the lengths of any spot formed by a scanlight on the target in the main scanning direction and the sub scanningdirection, and δ is the ratio of the smallest spot and the largest spoton the target.
 5. The two-element fθ lens of claim 1, wherein the ratioof the largest spot on the target and the smallest spot on the targetsatisfies the conditions of: $\begin{matrix}{{\eta_{\max} = {\frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}};} \\{{\eta_{\min} = {\frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}};}\end{matrix}$ wherein, S_(a0) and S_(b0) are the lengths of a spotformed by a scan light on a reflecting surface of the MEMS reflectingmirror in the main scanning direction and the sub scanning direction,and Sa and S_(b) are the lengths of any spot formed by a scan light onthe target in the main scanning direction and the sub scanningdirection, and η_(max) is the maximum ratio value of the largest spot onthe target with the spot on the reflecting surface of the MEMSreflecting mirror, and η_(min) is the minimum ratio value of the largestspot on the target with the spot on the reflecting surface of the MEMSreflecting mirror.